IV.—On Properties of Null Geodesics, and their Application to the Theory of Radiation

In a previous paper an expression was given by one of us for the Doppler effect in a de Sitter universe. In the present communication an attempt is made to obtain a more general expression for the Doppler effect, and for this purpose certain results have been derived relating to the properties of null geodesics in a general Riemannian space. In the paper referred to, the problem of defining spatial distance in general space was discussed and certain formulæ suggested which satisfy the essential conditions associated with the idea of spatial distance. This problem has now been approached from a different standpoint, and a new formula of more general form is now put forward. In the final section of the present paper a physical interpretation is given of the theorem relating to null geodesics which is proved in the first section. This involves a discussion of the concept of energy in general relativity, in order that a decision may be arrived at as to what law governs the transference of energy by radiation from one point to another. It appears that when the simplest and most natural assumptions are made for this purpose, then these are precisely the assumptions which are necessary to ensure the general consistency of relativity with the principles of quantum theory. This is illustrated by an application to a statical universe, and to the reddening of light in an expanding universe.